Primitive Heron triangles, ratios height/base or base/height is an integer

 

(A)

Find primitive Heron triangles whose base divides its height.

the height/base ratios 1 and 2 cannot occur,   but 3 can
 

for example,

the primitive heron triangle (4, 13, 15)
Area = 24,     perimeter = 32

base = 4,   edges = 13   and   15,   the height = 12

ratio height/base = 12/4 = 3

 
 

(B)

To find primitive Heron triangles whose height divides its base.

for example,
 

#1
primitive triangle (17, 113, 120)
area = 900,     perimeter = 250

base = 120,   edges = 17   and   113,   the height = 15

ratio base/height = 120/15 = 8

 

#2
primitive triangle (3389, 21029, 24360)
area = 7064400,     perimeter = 48778

base = 24360,  edges = 3389   and   21029,   the height = 580

ratio base/height = 24360/580 = 42

 

#3
primitive triangle (26921, 42041, 68880)
area = 56481600,     perimeter = 137842

base = 68880,   edges = 26921   and   42041,   the height = 1640

ratio base/height = 68880/1640 = 42

 

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